Can you do grade 9 math?

[Dark Hellion]

You know Bakustra, I went and I gave you the benefit of the doubt. It has been a while since I have done any work in philosophy of language (and that was very minor) or in formal mathematics so I decided to go do some light research. Now, I am going to need you to explain your whole "Cognition" angle in some detail, because I can not see the self-evidence you seem to think you have.

1) You're position of language acquisition and grammar is muddled to me. I have gone over your commentary quite a few times as I cannot determine what theoretical standpoint you are proceeding from other than a vague claim of cognitive science. There seems to be a nativist assumption that there is some innate grammar which we are simply labeling. Is this you position? If not, what is? It seems very disingenuous for you to dismiss the idea that grammar is artifice so out of hand given that it is a major competing theory to Universal grammar and Universal grammar has weighty criticisms laid against it.

2) Why do you assume that all the language mistakes of children are necessarily experimentation? Why is it impossible that they simply don't understand all the rules yet and while honestly trying to communicate are making a error in syntax? You even admit that adults can make mistakes in understanding complex sentences but some how children are immune to making actual mistakes in grammatical construction, its all just experimentation. Of course children are going to learn from the feedback they get when they make these mistakes, but this does not mean that the feedback was the motivation but can be an inadvertent consequence. There is a good deal of hypocrisy in you painting me as somehow upholding myself to be a master of psychology and yet making such overarching assumptions as to the motives of children.

3) The arbitrariness of math is also a subject of rather intense philosophical debate and not something to be lightly decided upon. Now, mathematics is quite surely a language, but to conflate it with the spoken languages in such a casual way is blatantly dishonest. Operators are not dependent upon external spoken language, as you imply, but are themselves a part of the language of mathematics. There is an underlying concept to the ideas that is not simply dependent upon the spoken language as you suggest but can be presented and represented in other completely analogous ways. 2+1=3 is an expression in the language of mathematics, but we can see what addition represents through a thought experiment such as "Now I have 3 cows. I had 2 cows and Bob gave me 1 cow." The addition is still present even though I used an entirely different language and changed the order. Furthermore, your theory of language dependent operators falls flat in the face of the fact that Chimps can be taught to conceptually perform (not memorize) simple math. So again I ask you to lay out this magical rebuttal you have wrapped up in the term "cognition".

4) About forgetting. Algebra is not a collection of facts that is simply memorized but a skill that is learned. Thus it will be forgotten in a similar manner to other skills such as the cliched analogy of riding a bike. There is an underlying level of understanding that is difficult to explain but once you have it, I mean really have it, just like riding a bike you never really forget, just get rusty. Now sure, there could be some point down the line that people forget it completely, but again this will be similar to other skills and probably be a very long time (anecdotally, I went 10 years without riding a bike without forgetting how). Look, Broomstick did it after 30 years of not using math. Now she may have an exception memory, but Mr. Wong's wife managed to do it after 21 years and others managed after a decade. Why is my expectation that people who really learned the skill should be able to remember it after a long time so absurd given the evidence we have seen? I never said I thought people should be able to do it perfectly without any outside references, in fact I specifically stated that someone may need to skim the chapter to remind themselves. You continuously constructed the strawman that I was expecting them to recall these problems without any preparation and with perfect clarity.

So, here is the deal. Point out were I lied about the contents of your posts. Point out this horrible crime that I have committed against you that I must apologize for.

[Bakustra]

You know Bakustra, I went and I gave you the benefit of the doubt. It has been a while since I have done any work in philosophy of language (and that was very minor) or in formal mathematics so I decided to go do some light research. Now, I am going to need you to explain your whole "Cognition" angle in some detail, because I can not see the self-evidence you seem to think you have.

1) You're position of language acquisition and grammar is muddled to me. I have gone over your commentary quite a few times as I cannot determine what theoretical standpoint you are proceeding from other than a vague claim of cognitive science. There seems to be a nativist assumption that there is some innate grammar which we are simply labeling. Is this you position? If not, what is? It seems very disingenuous for you to dismiss the idea that grammar is artifice so out of hand given that it is a major competing theory to Universal grammar and Universal grammar has weighty criticisms laid against it.

No, my position is that grammar is developed primarily through exposure to speech and language. Algebra, beyond the most basic levels of substitution, is not developed in such a way, and must be specifically taught later. Saying that grammar is artifice as a contrast to mathematics is a disingenuous position, because both of them are methods developed to understand that world and communicate that understanding to others

2) Why do you assume that all the language mistakes of children are necessarily experimentation? Why is it impossible that they simply don't understand all the rules yet and while honestly trying to communicate are making a error in syntax? You even admit that adults can make mistakes in understanding complex sentences but some how children are immune to making actual mistakes in grammatical construction, its all just experimentation. Of course children are going to learn from the feedback they get when they make these mistakes, but this does not mean that the feedback was the motivation but can be an inadvertent consequence. There is a good deal of hypocrisy in you painting me as somehow upholding myself to be a master of psychology and yet making such overarching assumptions as to the motives of children.

Because they are willing to speak despite the possibility of mistakes, so we can say that they are experimenting to understand the rules. There is no difference between my position and yours, save that mine explains why children talk rather than remaining silent to absorb grammatical rules.

3) The arbitrariness of math is also a subject of rather intense philosophical debate and not something to be lightly decided upon. Now, mathematics is quite surely a language, but to conflate it with the spoken languages in such a casual way is blatantly dishonest. Operators are not dependent upon external spoken language, as you imply, but are themselves a part of the language of mathematics. There is an underlying concept to the ideas that is not simply dependent upon the spoken language as you suggest but can be presented and represented in other completely analogous ways. 2+1=3 is an expression in the language of mathematics, but we can see what addition represents through a thought experiment such as "Now I have 3 cows. I had 2 cows and Bob gave me 1 cow." The addition is still present even though I used an entirely different language and changed the order. Furthermore, your theory of language dependent operators falls flat in the face of the fact that Chimps can be taught to conceptually perform (not memorize) simple math. So again I ask you to lay out this magical rebuttal you have wrapped up in the term "cognition".

Okay. Say "1" out loud. "One", right? That's the point. The operators are defined within our native language and we must understand them through that. If you could produce an expression that was incapable of being described via language, then you could reasonably argue that there was an independent "language of math" that people spoke. Otherwise, we can say that it is a matter of translation. Numerical systems are arbitrary. Is "1" any more representative than I or 一 or • of the initial positive integer? You can say that all these are similar, but consider the divergence between 20, XX, and 二十. So which is a more accurate representation? Are "+ and -" or "p and m" better representations of addition and subtraction? That is my point in saying that mathematics is no more arbitrary than language, because it is a language of its own, to which we assign arbitrary characters as its "letters"

4) About forgetting. Algebra is not a collection of facts that is simply memorized but a skill that is learned. Thus it will be forgotten in a similar manner to other skills such as the cliched analogy of riding a bike. There is an underlying level of understanding that is difficult to explain but once you have it, I mean really have it, just like riding a bike you never really forget, just get rusty. Now sure, there could be some point down the line that people forget it completely, but again this will be similar to other skills and probably be a very long time (anecdotally, I went 10 years without riding a bike without forgetting how). Look, Broomstick did it after 30 years of not using math. Now she may have an exception memory, but Mr. Wong's wife managed to do it after 21 years and others managed after a decade. Why is my expectation that people who really learned the skill should be able to remember it after a long time so absurd given the evidence we have seen? I never said I thought people should be able to do it perfectly without any outside references, in fact I specifically stated that someone may need to skim the chapter to remind themselves. You continuously constructed the strawman that I was expecting them to recall these problems without any preparation and with perfect clarity.

So then, why have you attacked me, if you have specifically stated that? Furthermore, prove it. Prove that factoring and simplification are things that can't be forgotten by people, and be sure to make it a general theory, rather than one based on a sample size of two. I have asked you to do so, and all you have provided is "It's like riding a bicycle! You never forget!" and taking a pair of anecdotes and turning them into a theory. But let's take a look at what you really said about algebra:

Of course you need to be familiar with the material and have practice with it to form a fundamental understanding, no one is making the pretense that you magically know this shit. The major point is that you should be familiar and have practice because you did this shit in high school. Math is like riding a bike, once you have actually learned how you just don't really forget it. Sure you may not be a quick as you used to and may not be able to ride no hands but you can get on the damn bike and pedal.

Hmm. I don't quite see where you said that. Let's see if you say this later on:

You still don't get it do you? All the rhetoric about off-hand is just obfuscation of the fact that either the people really couldn't do it or were too damn lazy to try to remember how and possibly make a mistake. It shouldn't be a question of memory excepting maybe the term factor. The idea that people can't see that simplify=make simple and isolate=make alone is mind-bogglingly stupid.

Not here either!

The entire test requires skills that are so basic that not knowing them is basically the same as not knowing Algebra and thus being incapable of doing Algebra. If you cannot reorganize terms in order to simplify or factor an equation and do this by fundamental understanding and not rote memorization you do not know how to do Algebra. This entire test is just this rearranging skill utilized over and over in different ways.

Nor here.

Now, I will readily concede that someone may not remember what the exact terminology means for factorization. Of course Zac Noloan asked what factorization was in the sixth post and was answered by D. Turtle in the ninth, only 97 minutes after the thread was created. Damn, all this inability to remember what factorizing means sure is interfering. And again, I can see how someone may make a mistake on what the form that simplification wants is. But neither of these are failures of applying the skill but simple misinterpretations of the directions. Look back at theJester for an example, when he checked his answer he quickly realized his mistake. I am willing to bet if we give him a new test he'd do just fine because he knows the concept.

This comes close, but not quite. So it seems that you are still a wee bit dishonest, Pinocchio. But feel free to point out where you specifically said that people might need to refresh their memory.

So, here is the deal. Point out were I lied about the contents of your posts. Point out this horrible crime that I have committed against you that I must apologize for.

So why is it impossible for somebody to have forgotten how to apply the skills, and therefore being unable to do them on a few seconds' or minutes' notice? You have not explained why you believe this to be the case.

You also appear to believe that you know what is going on inside other people's heads better than they do, but assuming that you are the master of psychology is again, befitting of your behavior throughout this thread.

Maybe it has to do with the fact that the skill is the fucking application for the most part? If I know how to FOIL I don't encounter a problem like (x+2)(x+3) and go, "well, I know I multiply the first, the outside, the inside, and then the last, but how do I multiply again?" If I know the interaction between exponents I need to do basic addition and subtraction. You are still using the false idea that knowing Algebra just meant being able to solve some problems for a test sometime back in the day. This is just cramming for a test. People who actually learn algebra by understand the groundwork will not have this application gap you are supposing because the primary thing they will have learned is how to apply the rules and theorems. This is the entire fucking point that you keep missing! The learning is in the ability to utilize and apply the knowledge, not simply the possession of it.

My question, and your lie are helpfully bolded. My question: "why is it impossible for someone to forget how to FOIL?" Your response: "It's impossible because learning how to FOIL means you can FOIL." In other words, it's either a matter of you being too dumb to understand what I am saying, or else blatantly twisting my words into a strawman easy for you to knock down, which, by the way, Pinocchio, happens to be lying. When I responded to this, you then ignored me, and that was what convinced me that you were being dishonest. So, then, Pinocchio, what shall be your response? I predict that you will handwave away or otherwise ignore my central question and focus on misinterpreting my statements and babbling pop-psych about the arbitrariness of math. Comprehension seems an unlikely possibility at this point.

[Rye]

Some aspects of it can be, to a degree. But not all. Factoring with exponents isn't intuitive if you don't remember the processes, for example.

I would tend to think that if you saw so much as one or two examples, it would come right back. Assuming you did well at it the first time around. For a lot of people, it might be something as simple as remembering what the word "factoring" refers to. We're still not talking about demanding some extraordinary level of recall, especially since no one said you couldn't refresh your memory before doing the questions. I have pointed out many times that this whole "march someone into a room and make him do a closed-book time-limited exam with no prep at all" thing is something others injected into the thread, not me.

To be fair, that's an entirely reasonable interpretation given that you posted some exam questions and asked if people could do them, in a thread called "Can you do grade 9 math". The implication there being that if you can't from the test questions alone, then no, you can't. Oh, and of course you should respect people less if they can't.

What do you mean by "base on"? Are you saying it should have zero effect whatsoever? And how do you respond to the charge that you and everyone else here is perfectly willing to make such judgements based on other kinds of cluelessness, albeit those which you find more "important" than math?

Are you trying to argue for an objective reason to dis/respect people? That seems absurd since respect is clearly a matter of personal aesthetics.

[ray245]

While I can do those question, I've made tons of careless mistakes. I do not have a problem with understanding maths and learning how to solve the questions, but I do have a problem with avoiding all the careless mistakes and handling the stress to finish all the questions on time.

[Dark Hellion]

So, lets see what we got from you Bakustra.

1) So after all your posturing about cognition you can only offer me vague allusion? Besides this you seem to contradict your previous position. I can bring it up if you want but earlier you did posit that grammar is simply the assigning of labels to already existent ideas and used this as refutation of the idea that grammar is about learning a construct.

Additionally, no where did I say math wasn't artifice (which is a hotly contested philosophical idea itself) but simply stated that it is no more constructed than grammar and seems to have some external connection that can be seen by the ability to formulate algebraic proofs in pictorial form or by utilizing real objects. You are simply ignoring a huge debate in the philosophy of math about how much mathematics is "developed" vs. how much is simply a representation of objective fact in order to maintain your position.

2) Perhaps children talk to communicate ideas? It is a radical concept that people might use speech to send and receive information but I am putting it out there. Even more outlandish is the idea the a child might be attempting to do this and fail because they lack certain language skills.

Nowhere did I dispute that children may learn grammar rules through feedback from their mistakes. But are you actually positing that this feedback is the primary motivating reason for children speaking and the erroneous syntax that can occur?

3) Again, failure to respond to my point. If language is so important to learning math why can we (who speak English) teach Chimps (who speak Chimp) to do basic addition? Why can computers perform math, despite not speaking any human language? How come I can draw algebraic proofs on a piece of graph paper and have you understand it, without using words? Perhaps there is a concept to operators that goes beyond simple words, a concept we have invented a language of math to communicate?

4) Algebra, like riding a bike is an example of tacit knowledge. Tacit knowledge is characterized by requiring a deep fundamental understanding that is not easily expressed in words. Tacit knowledge can be very resilient to memory loss, again the bike analogy or the fact that amnesiacs may not be able to remember their name but can play Beethoven on the piano.

And you didn't jump on Mr. Wong for saying the exact same thing as Ziggy and I said. Should I read into that?

Additionally:

It not like you are going to need a fucking night class to remember this stuff if you had ever bothered to learn it in the first place. You are going to need to skim the fucking chapter and look for the definitions.

Man, I wish I had specifically stated that some may need to skim the chapter and look up definitions. If I did, I would have tried to make it in my third post on the seventh page of the thread.

And now the last part. Here you show that you are really profoundly ignorant of the point that Ziggy, Mr. Wong, myself and others have repeatedly attempted to make throughout this thread. Now I will concede that my language was sloppy. I should have been more sure to specify that I meant the concept of FOILing (ie that one must multiply each term in the first parenthetical with all terms in the second and yadda yadda) and not simply the algorithm of FOILing. This is what people have been trying to get through in this thread for several pages. Algebra is not about memorization of algorithms and terminology but is about learning and understanding the fundamental ways numbers and variables interact and can be interacted with. This is exactly why I say that the skill is in the application. It is not about knowing the algorithm but about knowing why we would use that algorithm and when we would use it. This is why I demonstrated that one can develop the FOIL algorithm via the use of substitution or that one can generate the general rule for exponential interactions via substitution. Simplification and factorization are ultimately reducible to simple logic and arithmetic operations. Your entire application caveat that you are trying to shove through is a red herring.

By the way, Pinocchio is not as effective of an insult as you think it is. At the conclusion I get to kick a giant whale's ass and become a real boy, in the end you are still a hypocrite.

[Spoonist]

Sorry that I didn't have time to respond properly before. This means that I'll have to skip some of the middle arguments and head straight for the ones at the end. If anyone thinks that I should adress a specific middle argument please send me a PM.

So lets see where this discussion evolved.

Now since Darth Wong seems to have forgotten how the 'respect' part started here is his quotes from p2 just as a backdrop.

I take it you have no kids? Peoples' lousy math skills often represent the beginning of their childrens' loss of respect for them. If you're 13 years old and your parents are stumped by your math homework, that's not going to improve your opinion of them.

Quite frankly, it's a pitiful self-justification to say "Oh well, I suck at math but I'm good at English". Math is the language of logic; if you suck at math, it means your logic skills are poor. It's not really a compensation to say "Yeah, but I can write a mean Shakespeare literary analysis essay".

Is it correct that you still think that those two quotes are true? Is it correct to assume that even though you have changed your argument since, that you still hold the view that someone who is self proclaimed sucky at math, regardless of other academic achievements, should be considered to be a less logical person?

So I think less of people who suck at math, and I don't make some holy exception for parents. Ooooh! That's totally different than judging people for having right-wing militarist opinions, or George W. Bush for forgetting his geography, or pronouncing "nuclear" wrong!

Yes. That is totally different.
a) most adults suck at "understanding" math for practical implications, that is why social pyramid schemes, credit cards, the shadier parts of the loaning industry, get the phone for free with a service, etc all work as business concepts.
b) If the parent has brought up the kid properly & lovingly then the kid usually respects that parent regardless of any lack of knowledge they have, to the point of inheriting ignorant political and religius views
c) A person advocating something as complex as a specific political opinion tells a lot more about their personality than their own perceived notion of how good they are at math.
d) W was never judged on his geography or pronounciation skills alone, but in the context of his whole persona and his position as seeking and getting one of the most powerful positions on earth it became a standing joke, had he had a different position like janitor that had not been funny
e) What is missing in your examples is something else that would relate to a 13 year old. Since they usually don't grasp the full implications of a political viewpoint or the president making lots and lots of tiny educational mistakes
f) Where do you draw the line for when to "think less of" people for their self proclaimed sucky math skills? high school, college, university? Which grade average should they have? For how long should they retain this knowledge?

The problem is that algebraic rearrangement is so very, very simple. As I said, it's so simple that it's part of the curriculum before math becomes option in high school, and for good reason: if you understand the concept of a variable, what equations mean, what plus and minus and equals are, then it's fucking obvious.

Its simple to people like you and me who "get math", its not simple to the majority of people. Otherwise grading math in the 9th grade would be real easy, just straight As and no failures. But that is not consistent with reality. You yourself said it yourself in the beginning, most people think that math is hard and its one of the hardest subjects to bullshit your way through.
Now you can not have it both ways, either its so simple that any kid can do it and should be considered a simpleton if they cant. Or it is hard and difficult to bullshit. Now the truth of course is a grayscale somewhere in between. Its easy for some, hard for others and there are some who could bullshit their way through by rote memorization.

It's not a matter of memorizing things.

Not this again. Are you still saying that a 9th grader can't get through math by cramming alone? Because that is counter to what you yourself are saying elsewhere. People who struggled the first time through it and barely passed to then forget all about it after the summer break are all living proof that you can do this. As I pointed out earlier you can even do this in engineering school if you are smart enough.

Would it seem odd for you to insult someone who thinks the Nazis could have won WW2 if they just did one or two things differently? Because that has happened around here on numerous occasions, and what is that? Attacking someone for something that is not necessary for 99.9% of people in their jobs.

That's is quite a dishonest comparison. You are equating real life relationships like parent-child with the antics of a very pecualiar web forum. Of course its not odd in the context of the history forum to insult someone for their ignorant WWII opinions, that is one of the specificly stated objectives of that forum. Would it be odd to act like we do here in real life? Of course it would.
Your examples shows your own bias perfectly.

Someone who can't rearrange an equation, even after 20 years, must have been utterly clueless the first time around. This is a point that's been made again and again in this thread, by myself and many others.

And has been argued against again and again. Especially since you again say "equation" as a generic term and not "9th grade equations", which includes some pretty complex stuff.
As I said before, I help my engineer colleagues with their kids' homework. This is not due to them being clueless the first time round, rather the opposite. For the smartest its hard for them not to overcomplicate things, trying to solve stuff that is not supposed to be solved, or, trying to teach the kid "why" the formula works that way instead of "how", or, trying to use a higher level shortcut making it impossible to solve the middle steps. Then there are those engineers who simply never 'liked' things like variables in math and promptly glaze over when they see them.
Agreed, that it usually just takes 10-15 min before they get it but it still makes your categorical assertion wrong.

[Spoonist]

Some small observations from when I was tutering 16 year olds in math to make some extra cash.
1) Rote memorization is an essesntial step in learning a math concept. This is why one does the examples over and over again. However most teachers fail to impart the knowledge that it is only a step and not how one should approach math. So most students are consistently taught the wrong approach.
2) A key element for the struggling students is confidence and self perceived knowledge. If they think they suck at math and that they will fail, that will become true. Reversing that self perception can be more important than actually learning the stuff.
3) Usually math teachers who are good at math, suck at teaching it to the majority of the students. This because they "understand it" and fail to comprehend why others could not make the same leaps in logic. So their explanations only make sense to the top percentile.
4) Most math concepts is built on a former concept. So if you take lots of short cuts, like cramming, it will become progressively harder to do so. Sometimes just showing the students how much would come back to haunt them in later courses would make them put in some extra effort to understand the concept.
5) Most good looking girls did not "want to" do well in math (unconsciously). This because of the prejudice that good looking=bad at math. So they could be really smart, solve all the exercises easily and then fail tests.
Some small observations from my engineer colleagues. (I work for one of the larger engineering companies in the world).
1) Most of them do not enjoy math. This translates into an unwillingness to spend time on it. Thus leading to them forgetting basic stuff. This even if they at the same time do really advanced math-based stuff.
2) Most do not perceive that what they are doing is math-based even when it is. This because they have bought the meme that math is boring, so anything that is fun can not be math.
3) Those who are the best at the realy super advanced math-based stuff, geniuses well beyond my reach, also sometimes have problems with explaining stuff to their kids. This because the middle steps are too simple. So they also come to me asking how to explain stuff.

[Bakustra]

So, lets see what we got from you Bakustra.

1) So after all your posturing about cognition you can only offer me vague allusion? Besides this you seem to contradict your previous position. I can bring it up if you want but earlier you did posit that grammar is simply the assigning of labels to already existent ideas and used this as refutation of the idea that grammar is about learning a construct.

Additionally, no where did I say math wasn't artifice (which is a hotly contested philosophical idea itself) but simply stated that it is no more constructed than grammar and seems to have some external connection that can be seen by the ability to formulate algebraic proofs in pictorial form or by utilizing real objects. You are simply ignoring a huge debate in the philosophy of math about how much mathematics is "developed" vs. how much is simply a representation of objective fact in order to maintain your position.

I am talking about language in general. The concept of a subject, verb tense, or other part of grammar is something that children will learn before being formally taught the definition of, seeing as five- and six-year-olds can construct simple sentences. But with language in general, a word is assigned to a concept, and math is represented in terms of language. Consider varying bases and you will see why the idea that math is developed is not actually contradictory to the idea that math represents objective pact.

2) Perhaps children talk to communicate ideas? It is a radical concept that people might use speech to send and receive information but I am putting it out there. Even more outlandish is the idea the a child might be attempting to do this and fail because they lack certain language skills.

Nowhere did I dispute that children may learn grammar rules through feedback from their mistakes. But are you actually positing that this feedback is the primary motivating reason for children speaking and the erroneous syntax that can occur?

So in your world, children only communicate verbally? Besides, I would like to see what sort of if/then statement is something that children would try to make if, as you stated, they cannot actually comprehend an if/then statement. For that matter, explain the rise of Nicaraguan Sign Language, if children don't experiment with language and grammar.

Something can have more than one purpose, and attempting to use unfamiliar concepts is the only means young children have to test whether their understanding of them is correct or not; but many unfamiliar concepts are ones that they don't use regularly (with the possible exception of "to be" and other common irregulars).

3) Again, failure to respond to my point. If language is so important to learning math why can we (who speak English) teach Chimps (who speak Chimp) to do basic addition? Why can computers perform math, despite not speaking any human language? How come I can draw algebraic proofs on a piece of graph paper and have you understand it, without using words? Perhaps there is a concept to operators that goes beyond simple words, a concept we have invented a language of math to communicate?

How do you program a computer to perform addition? You translate the steps necessary to perform addition into a programming language, which is itself translated into machine code by the computer and then executed. So, yes, language is fundamental to understanding math. We can teach by example, but the chimpanzees translate the concept into their own language (assuming they have one) and algebraic proofs also rely on language. If there was true universality to math, no one would be incapable of understanding it, yet the language of math must be taught. So it is not universal. It must be understood by people within their own language as well, at least initially. But you are right that there is a concept to operators that goes beyond words, but so too is there a concept to octopuses that goes beyond words.

4) Algebra, like riding a bike is an example of tacit knowledge. Tacit knowledge is characterized by requiring a deep fundamental understanding that is not easily expressed in words. Tacit knowledge can be very resilient to memory loss, again the bike analogy or the fact that amnesiacs may not be able to remember their name but can play Beethoven on the piano.

Prove it. You still have dodged the question and refused to provide a source.

And you didn't jump on Mr. Wong for saying the exact same thing as Ziggy and I said. Should I read into that?

Yes, but the lesson you should take away is that Mr. Wong responded to me civilly and I responded to him civilly. You and Ziggy responded to me rudely, and I responded rudely as well. Your insinuations fall flat.

Additionally:

It not like you are going to need a fucking night class to remember this stuff if you had ever bothered to learn it in the first place. You are going to need to skim the fucking chapter and look for the definitions.

Man, I wish I had specifically stated that some may need to skim the chapter and look up definitions. If I did, I would have tried to make it in my third post on the seventh page of the thread.

Man, I sure wish that you would read my posts, since it seems that we don't disagree on this point and you're just being pissy because I dared to disagree with you.

And now the last part. Here you show that you are really profoundly ignorant of the point that Ziggy, Mr. Wong, myself and others have repeatedly attempted to make throughout this thread. Now I will concede that my language was sloppy. I should have been more sure to specify that I meant the concept of FOILing (ie that one must multiply each term in the first parenthetical with all terms in the second and yadda yadda) and not simply the algorithm of FOILing. This is what people have been trying to get through in this thread for several pages. Algebra is not about memorization of algorithms and terminology but is about learning and understanding the fundamental ways numbers and variables interact and can be interacted with. This is exactly why I say that the skill is in the application. It is not about knowing the algorithm but about knowing why we would use that algorithm and when we would use it. This is why I demonstrated that one can develop the FOIL algorithm via the use of substitution or that one can generate the general rule for exponential interactions via substitution. Simplification and factorization are ultimately reducible to simple logic and arithmetic operations. Your entire application caveat that you are trying to shove through is a red herring.

So your entire argument is that somehow, "forgetting the application" means something other than forgetting where and when to apply FOILing (the application), but rather means forgetting the actual algorithm. Yes, well, I will retract and apologize for saying that you were dishonest, since apparently you are far dumber than I had thought. I'm sorry for assuming that you knew what "application" meant.

By the way, Pinocchio is not as effective of an insult as you think it is. At the conclusion I get to kick a giant whale's ass and become a real boy, in the end you are still a hypocrite.

Responding to insults in this fashion does not actually do what you think it would do, buddy. (The general aura is pathetic rather than triumphant, just as an example).

[Dark Hellion]

You know Bakustra, after reading your rebuttals I really just wanted to reply and just tear into you. But after a nap and a short walk I realized that that wasn't necessary. Simply highlighting the ignorance and stupidity of your responses will be more illustrative and damning than any insults I can level. In fact, I am not even going to use profanity (not even my favorite adjective/adverb fucking) because I don't think there has actually been a curse word invented that can convey just how idiotic some of the things you said are.

First, since it will be illuminating to you, I am going to construct all the Algebra to do this assignment from very simple principles. Note that this is a practical exercise and not meant to be a formal treatment but I think is should be sufficient. This is going to be rather large but should be much more compact than the implications being leveled against Algebra would suggest.

Let us start with a group of counting numbers. We will start by defining them purely pictorially. (.),(..),(...), etc. To these we will assign Arabic numerals so they become 1,2,3,4, and so on until we hit some arbitrarily high number we will simply call many. Finally, let us have one final number called zero, pictorially () and in Arabic 0 (I'll admit to some intellectual laziness in my usage of zero here but this is practical and zero has 1300 or so years of formal development for us to draw off). I will finally define a symbol (=) to represent a relationship that means that the contents of one side are equivalent to the other. From here I can get general identity in that (....)=(....) but can also easily go further.

Now I am going to define an operator that will be our groundwork for addition. However, this will not be a binary operation like normal addition. I am simply going to define an unary operator ([+]) that increments the number up one. This operator is not necessarily mathematical, it is a mechanical instruction to place an additional dot in our pictorial form. So (..)[+] will become (...) or we can say (..)[+]=(...). We can rewrite this in numerals as 2[+]=3 which is a translation of the dot representation. From now on I will use the Arabic numbers but it should be obvious that this is for space conservation and that there is still a deeper mathematics going on that is only utilizing the dots, the basic logical notion of equivalence and as will be seen this purely mechanical unary operator.

Now I will develop a more advanced addition that will eventually be represented as (+). First, let us notice that any number can be generated by successive applications of the mechanical [+] operator upon zero. 0[+]=1, 0[+][+]=2, and so on. So let us define addition as a binary operator which tells us to apply the [+] operator to the first number for the second number of times. So, 1+2 is actually 1[+][+]. We can see that 1+2 = 1[+][+] =3. From here we can get commutativity by noting that 1+2=3 and 2+1=3 and can demonstrate this ad infinitum until satisfied. We can demonstrate additive identity by noting that 0+2=2 and by commutation 2+0=2. We can see that numbers can be decomposed by reversing the order of an already known equality such as 3=1+2. We can utilize these developments to get associativity as well.

We should pause for a second and examine what we have done so far. We have built an addition operator that is complete enough for the practical usages we would encounter in everyday life. To do so our foundation has not been in any spoken language, that is just for communicating the concepts to you. Our foundation is actually in dots and a mechanical operation that has us draw an additional dot and the very basic logical concept of equivalence. All the symbols I have assigned are purely arbitrary and could just as easily be Chinese characters, wingdings or tiny pictures of mecha.

OK, now I am going to notice that there is another way we can go. We could decrement one instead, mechanically erasing a dot. This will be [-]. But quickly we run into a snaggle. What happens if we have 0[-]? We seem at an impasse so lets just make something up to stick in there and start to play and see what we get. We will say that 0[-]=()[-] will become (.) and we will call this -1. We will now look at [-] as decrementing black dots until we get to zero and then adding red dots. We should note that this is just an assumption and we have defined very few properties of negative numbers at this point. For now we will simply say that a number of red dots is represented by the counting of those dots and the placing of a negative sign in front of that quantity; that is -2=0[-][-]. From here we can reproduce the concept by which we developed addition to get a general binary operator for subtraction (-), finding basic examples like 2-1=1 or 1-3=-2 and explore the properties of this operator. More importantly though, we can note that [-] is a reversal of [+] and notice the concepts we can draw from this. We can see by simple mechanics that if we take 3[-][+] we get 3; that a [-] will cancel a [+] and by extension a black dot will cancel a red dot or the opposite. We can see that [+] can now be considered to take away red dots until we have no more dots and then add black dots. Now we can figure out how negative numbers interact with addition. If we want to take -1+2=? we can solve for the question mark. We represent -1 as 0[-] and convert the binary addition back to our unary operator and see -1+2=0[-][+][+]=1. We can commute this because it is addition and get 2+ (-1)=1 and then compare this to 2-1=1 and notice that 2-1=2+ (-1). We can repeat this until we are satisfied that subtracting a number can be seen as adding its negative and vice versa. We can see that the addition of two negative numbers gets us another negative such as -1+ (-1)= 0[-][-]=-2 and observe other interactions. We now have an explorable group of integers and yet at our very base we are still just using colored dots which we mechanically draw or erase.

Continuing we can notice that on occasion we may want to add the same number to itself over and over again. For example 3+3+3+3 or -4+ (-4). We would like a shorthand so will define a new binary operator (*) that tells us to add the first number to itself a number of times equal to the second. Thus our first example becomes 3*4 and our second example becomes -4*2. We could also view this as having a number of groups each containing a specified number of dots. We run into a small problem with either of these views in the question what is 4*(-2)? Luckily, if we play around with well defined examples of multiplication we can convince ourselves of both the commutative and associative properties of multiplication. Now 4*(-2) can be seen as (-2)*4. We can even see how to multiply two negatives such as -2*(-2) by using a function we could solve such as -2*2*(-2). By association we can see that this is equivalent to -2*(-2)*2 and must equal 8 as well. Repetition shows that any time we multiply two negatives we will have a positive. We can extend this further to an assumption that multiplying by a negative should reverse the sign of the other number, which suffices for our practical demonstration. We can also note that 1 is a multiplicative identity and that any number times 1 is just the number. Finally, we can again note an aspect of decomposition in that 16=4*4=2*8=2*2*4=2*2*2*2.

Now before we proceed further I think it would be prudent to introduce the concept of a variable. Going back to our dot picture we will imagine a delineated space which we know contains some unknown but static number of dots. For identification purposes we will define three spaces as a triangle, a circle and a square but these need only be any distinctive enclosed space, the shape is unimportant. We will propose that if the shape is written in black that we should simply consider the number of dots inside, if it is red we should consider a number of dots of opposite color. We will assign x,y, and z for the black triangle, circle and square respectively and -x,-y, and -z for the red counterparts. Trivially, we can see that x[+] is to add one black dot to the number of dots within x (remembering that black dots cancel red dots). We can further see that x+2 will add two dots the amount in x or that x*2 (which from now on will be understood as 2x=2*x=2(x)) will give us two triangles with equal numbers of dots. We can now ask the question, what if we have an equality like x+4=7? We can see that x contains a number of dots such that if we add 4 more dots we have a total of 7 dots. Now we hammer at this with logic for a bit; how many dots does it take to go from the 4 dots we know we have to the 7 dots we end up with? We can easily count this on our fingers and get 3. But how would we solve for a general problem? Let us make a minor intuitive leap by noticing that by our conceptual definition the two sides of the equality must be equivalent and that as long as we act symmetrically it will remain so. We can also refer back to our concept of decomposition and see through limited trial and error that 7=3+4. Noticing that we can represent x+4=7=3+4 as x[+][+][+][+]=3[+][+][+][+] we can see if we symmetrically apply four [-] operations to each side we will cancel all the [+] and get x=3 which checks from our basic counting. Continuously applying this to problems involving single variables and only addition and subtraction we can see that we need to isolate our variable on one side by applying the inverse operations to get an equality for what the variable's value is. We should also note that if we are given a definition of the number represented by the variable we can readily substitute it. We can further expand this by noting that a variable is just an unknown number of dots just as an Arabic number represents a known number of dots and can move the variable around by using inverse operations. We can now gain equalities from such problems as x+5-y=3 and can define x in terms of y as x=y-2 or y in terms of x as y=x+2. Unfortunately we cannot solve 2x=4 with this alone. We now need an inverse of multiplication.

So, lets explore what the inverse of multiplication would be. If multiplication is the grouping of similar elements into a larger whole then its inverse would be to divide that whole into a number of similar elements. We will call this operation (/) and take it as a binary operation were we take the number in front and divide it into the second number of equal groups. We begin in the most basic method possible, just taking symmetrically arranged dots and bisecting and then counting the dots in each equal group. Thus 4/2 will become (..)|(..) or two groups of (..), 4/2=2. To divide by 3 we would divide a number into 3 groups such as 9/3 is (...)|(...)|(...) or 3. Finally, we exhaust the trivial by noting that a number divided by itself will get us a number of one dot groups equal to our initial number of dots and thus be one. Compared to our previous arithmetic operations division seems much more muddled in how we quickly gain a solution. Primarily it seems we should rely on the decomposition we see in multiplication. By noting that a number like 54=3(3)(3)(2) we can see that 54/3=3(3)(2)=18 via our understanding of the functioning of inverses. This is satisfactory to solve a problem such as 2x=4. We divide each side by 2 and get x=2. We can further see that x/x=1 from our knowledge about variables and multiplicative identity. Lastly we can notice that there is a problem with x/0, by inversion there is no number we could multiply by zero that would give us a non-zero number. Additionally for the purpose of 4b we should note that we can have something like x=7/3 taking it as the implication that we need to complete the operation.

One operation to go. On occasion we may even find ourselves wanting to multiply a number by itself a number of times, that is to repeatedly form groups of groups. We will define this operator as the exponentiation operator (^) and say that we multiply the first number by itself the second number of times. Thus 3^3 is 3*3*3=27. Pictorially this is three groups of three groups of three. 2^4=2(2)(2)(2)=16 or two groups of two groups of two groups of two. We can see by definition that x^4=x(x)(x)(x) or x groups of x groups of x groups of x. It should be obvious that we could set an equality like x^3=64 and solve it with an inverse operation but this is unnecessary for the assignment we are given. Instead let us examine how exponentiated variables interact under the arithmetic operations. Lets just take a simple form of x+x^2. We wonder if we can combine these x's in some way. Let us insert a number for x, say x=3. We would have 3+3^2=3+3(3) which requires us to have an order of operations to solve. However, our concept building has already given us this. We break each operator down into the simpler operator it is based off of. We start with exponents because they are based on multiplication which is based on addition which is based on our reversible mechanical unitary operators. To break it down completely into a totally obnoxious form we have 0([+][+][+])([+][+][+])([+][+][+])[+][+][+] and this gives us our expected answer of 12. So the order of operation implied by our very original mechanical operator shows us we must resolve exponents before we add and thus x+x^2 is as simple as we can get. Now what about multiplication and exponents. Let us take x^2(x^4); we'll begin by breaking it down into (x)(x)(x)(x)(x)(x) and notice that this is x^6. Repeating for many different combination of exponents and we observe (x^y)(x^z)=x^(y+z). Now division, x^4/x^2 is x(x)(x)(x)/x(x) and by canceling we get x^2. Repeat and we get (x^y)/(x^z)=x^(y-z). Now we finally examine one last interesting case of x^0, which we set as x^2/x^2 = x(x)/x(x) =1 and will for any other combination of x^y.

We have derived everything to solve all problems but number 2 and 3b. We need to examine how to multiply a mixture of variables and defined numbers. We again start in the simplest way be examining an equation like (x+4)(x+4)=(x+4)^2. Let us just suppose that x=5. We revert to our original operator, taking 5+4 and then square that. We have 81. Now examining what we had originally, we see that we put at least a 5^2 and a 4^2 into 81. Subtracting those from 81 we find that 40 remains. So now lets just play by multiplicatively decomposing 40 (we could term this as factoring as well but I am using a more general colloquial term on purpose) and we see that we get 4(5)(2). Hmm... we have our 4, our value we assigned to x and 2 left over. Going back and generalizing we see that (x+4)^2=x^2 + 2(4)(x) + 4^2. We will see this holds for any assigned value of x. In fact, we can go back and generalize more and get (x+y)^2=x^2+2xy+y^2. So lets go to the next step; we start with (x+y)(x+y) and set y=z so that we'll have (x+y)(x+z)=(x+y)^2. Looking at it this we can see that we could represent our general answer as x(x)+xy+xz+yz which equals x^2+2xy+y^2 by substitution. We can see that we must multiply each element in one portion by the all elements of the other portion and add all these chunks together. We can invent an algorithm for this (the FOIL method) but the concept is just a natural extension of our previous rules. We can see that this process is reversable, that x^2+xy+xz+yz can be reverted to (x+y)(x+z). We can test this repeatedly and see it holds in general. Further experimentation can convince that we can use portions of any size and of dissimilar sizes as long as we remember to multiply each element and add.

Whew... that was quite a workout. Now, I will note that this is not a very rigorous system, the properties aren't really given by proof but by a general satisfactory repetition. But the amazing thing is that this is entirely independent of any spoken language at the conceptual stage. One could translate these instructions into any language and the base concepts are totally unchanged. The entire thing is built from objective properties about discrete objects, a reversible mechanical action and a division of space. The dots could be cows on a farm, variables barns containing cows, our unary addition is a cow arriving at the farm, unary subtraction is a cow leaving the farm, multiplication is combining multiple pens of cattle, division is splitting a herd into pens, and exponents are conglomerating groups of groups into a giant herd. In fact, all Algebra excepting variables raised to exponents, or the multiplication of two variables can be performed purely mechanically in this system. {(4^2)(x)+4}/2 can be done with paper triangles and Othello pieces. Now x^2 or xy requires us to actually use a symbolic notation, but this notation is arbitrary and x^2 could easily be represented with and is just as meaningful if we maintain the conceptual connection back to our dot/space framework.

This is the profound fundamental nature of Algebra that you have been missing Bakustra. I have derived the rules for simplification entirely from a relationship between dots on a paper, arbitrarily chosen spaces that hide dots and the desire to have less symbols written on a different piece of paper. I have derived factoring from basic Algebraic substitution, general equality, and trial and error with a known function. None of this relied on any language outside the mathematical language I developed. Any and all terminology could be replaced with total nonsense words or crudely drawn pictures and it still functions. Multiplication could be called "glurbalglobali" and it doesn't matter because it is defined off addition which is defined off the incrementing operator which is defined by the mechanical act of drawing a dot.

Now I guess I should rebut your "points".
1) You brought up cognition, I asked you a question on it, now answer the question. Do you think grammar is innate or is it a construct? How are you defining syntactic development? What about the very relevant semantic issue that words have context based definitions where a word can represent an entirely different concept if a situation is changed? Will you address any of these or keep trying to weasel out and deflect with vague claims of cognition?

2) Stop the strawmanning. I never said children cannot form if/then statements. I never said children don't use non-verbal communication. I never said children don't experiment with language. I think it is foolish to assign this as the primary motivation for children to speak. Children can hold conversations, they can ask for things, make demands, convey emotions. These are not being done for the express purpose of experimenting with language; the learning is a secondary effect. And you are blatantly dismissing the effects of other forms of negative and positive reinforcement. Children can learn from observing the grammatically correct speech of others and from the absence of certain seemingly grammatical structures in this speech.

3) Where do I start? The equivocation between the concept of addition and the concept of an octopus? How about the idea that a drawing can rely on language. Is Chinese graph paper harder to read if you are a native English speaker? Maybe I should examine the point on Chimps. You parenthetically admit that Chimps might not have a language and still insist language is fundamental to math. You managed to do a complete 180 in under a sentence. And even if Chimps do have a language are you arguing that they have the necessary semantic and syntactic components to construct a workable description of addition? And that they can understand English well enough to translate our explanation of addition (which is fundamentally based on language by your argument) into Chimp?

No, I am going to go with the machine language. You realize that add is a basic component of machine language right? A computer does not translate other commands or instructions into addition, addition is one of the basic functions from which it makes more complex computations. A computer is a dumb box of rocks that happens to be really fast. Addition is handled via physical properties of the hardware on command from the machine language. The addition is not based on the language, the language is based on the fact that the machine can add. This is the exact opposite of what you are trying to pass off.

4) It's only wikipedia but I would think the mass of anecdotal support for this exact position should suffice. I have a feeling you'll provide some flailing excuse.

5) On application. I am tired of fighting this false distinction you keep putting up. The skills of Algebraic reorganization and substitution are primarily concerned with the application of underlying mathematical concepts. This is why I have said that forgetting the application is the same as forgetting the skill, because they are skills of application, not algorithms or operations to be worked through but understanding how we can apply arithmetic rules. This is the thing you are seemingly unable to understand. Its like a physicist saying he knows the right hand rule for E&M but doesn't remember how to apply it or for a math example it is saying you know the rules of multiplication but don't know how to apply them. Could I say I know that multiplication is commutative if I don't realize I can switch the order of multiples involved? This is why people keeping using phrases like "really learned" to denote the difference between memorizing the words behind a concept and actually being able to understand the concept.

[Channel72]

This is the profound fundamental nature of Algebra that you have been missing Bakustra. I have derived the rules for simplification entirely from a relationship between dots on a paper, arbitrarily chosen spaces that hide dots and the desire to have less symbols written on a different piece of paper. I have derived factoring from basic Algebraic substitution, general equality, and trial and error with a known function. None of this relied on any language outside the mathematical language I developed. Any and all terminology could be replaced with total nonsense words or crudely drawn pictures and it still functions. Multiplication could be called "glurbalglobali" and it doesn't matter because it is defined off addition which is defined off the incrementing operator which is defined by the mechanical act of drawing a dot.

Your ability to derive all of this from first principles is admirable, but if algebra is as intuitive as you claim then how come a human understanding of arithmetic existed for millenia before al Khwarizmi finally formalized the idea of combining like terms in a generic equation? Mankind has been capable of simple arithmetic for millenia, but only a few geniuses in the Hellenistic period ever considered algebraic concepts. It wasn't until the Middle Ages, after ~3,000 years of human civilization, that al Khwarizmi formalized the study of generic equations which eventually came to be called "algebra." So your expectation that the average person should be able to derive algebraic concepts from first principles based only on their knowledge of arithmetic is utter bullshit.

Let's face it, the mathematical skill of the average human being is generally limited to basic arithmetic with natural numbers. Most pre-agricultural hunter-gatherer tribes didn't even have words in their language for numbers higher than 5, let alone zero or fractions. We may be in the 21st century, but the human brain hasn't changed. You can't reasonably expect the average adult to be able to derive algebraic concepts from first principles without some kind of training or reference material.

[Bakustra]

*snip*
This is the profound fundamental nature of Algebra that you have been missing Bakustra. I have derived the rules for simplification entirely from a relationship between dots on a paper, arbitrarily chosen spaces that hide dots and the desire to have less symbols written on a different piece of paper. I have derived factoring from basic Algebraic substitution, general equality, and trial and error with a known function. None of this relied on any language outside the mathematical language I developed. Any and all terminology could be replaced with total nonsense words or crudely drawn pictures and it still functions. Multiplication could be called "glurbalglobali" and it doesn't matter because it is defined off addition which is defined off the incrementing operator which is defined by the mechanical act of drawing a dot.

You really don't understand what I am saying, do you? When you provide a symbol for the operation, and then demonstrate it, then two things are happening: 1) you are producing a written shorthand for the operation and 2) the person you are demonstrating this to is assigning a name to the operation internally. While you can call multiplication anything you want, you can't give it the same name as addition, and you couldn't use the same symbol for both either. So yes, language is important, because even an invented mathematical language will still be translated by the recipient into his own language (assigning names if necessary) and it still needs to follow basic rules to be useful.

Now I guess I should rebut your "points".
1) You brought up cognition, I asked you a question on it, now answer the question. Do you think grammar is innate or is it a construct? How are you defining syntactic development? What about the very relevant semantic issue that words have context based definitions where a word can represent an entirely different concept if a situation is changed? Will you address any of these or keep trying to weasel out and deflect with vague claims of cognition?

I am of the opinion that the basics of language are innate in humanity (see Nicaraguan Sign Language for why) but that grammar is at least partly constructed. Meanwhile, what do you mean, "how am I defining syntactic development?" The definition of syntactic development is literally the development of understanding of syntax. If you meant to ask what model I find convincing, that would be, in general, the error-recovery model, seeing as I presented it in the thread. As for your "relevant semantic issue", I'm not sure why you think that my statement denies the possibility of multiple meanings, seeing as all I said was "a word is a assigned to a concept".

2) Stop the strawmanning. I never said children cannot form if/then statements. I never said children don't use non-verbal communication. I never said children don't experiment with language. I think it is foolish to assign this as the primary motivation for children to speak. Children can hold conversations, they can ask for things, make demands, convey emotions. These are not being done for the express purpose of experimenting with language; the learning is a secondary effect. And you are blatantly dismissing the effects of other forms of negative and positive reinforcement. Children can learn from observing the grammatically correct speech of others and from the absence of certain seemingly grammatical structures in this speech.

But this is not a model that I developed; there have been several models based on error-recovery seriously proposed, and unless you have some means of discrediting these models or supporting your claims, then I think it is somewhat premature of you to declare that my difference of opinion in this case is a sign of idiocy. Of course, I was too hasty, and I withdraw any claim that experimentation is the only means of obtaining grammar.

3) Where do I start? The equivocation between the concept of addition and the concept of an octopus? How about the idea that a drawing can rely on language. Is Chinese graph paper harder to read if you are a native English speaker? Maybe I should examine the point on Chimps. You parenthetically admit that Chimps might not have a language and still insist language is fundamental to math. You managed to do a complete 180 in under a sentence. And even if Chimps do have a language are you arguing that they have the necessary semantic and syntactic components to construct a workable description of addition? And that they can understand English well enough to translate our explanation of addition (which is fundamentally based on language by your argument) into Chimp?

While a line is a line, the notations for the graph are going to differ, and so would the numbers if the world hadn't adopted Arabic numerals in general. However, the point is that an octopus and addition are both concepts that are represented by words, at least internally if nothing else. Unless you have some means of conveying addition that ensures that nobody involved invents a mental definition for addition, then you have little ground to stand on in arguing that language is completely distinct from math.

When it comes to the matter of chimpanzees, while I personally believe that they are capable of learning language and may well have one of their own, I decided to avoid offending Chomskyites. My argument is that a chimp that is taught addition will have an internal description of the parts of addition, as well as a distinguishing method from subtraction, multiplication, etc. Even if this is not a language per se, they can still distinguish between individual chimpanzees, which suggests a proto-language and the ability to define objects. Further, your argument is a strawman. They translate the numerical forms of addition into definitions/words of their own, by my argument, so that a chimp will invent a word for "add" or "subtract" to distinguish it from the other methods it has learned, as well as being able to define the numbers and signs.

No, I am going to go with the machine language. You realize that add is a basic component of machine language right? A computer does not translate other commands or instructions into addition, addition is one of the basic functions from which it makes more complex computations. A computer is a dumb box of rocks that happens to be really fast. Addition is handled via physical properties of the hardware on command from the machine language. The addition is not based on the language, the language is based on the fact that the machine can add. This is the exact opposite of what you are trying to pass off.

Machine language is still a language that tells the computer to do this or that with its hardware. "Addition" in most machine languages consists of single increments or decrements, so, no, machine language does not incorporate addition beyond the most basic level.

4) It's only wikipedia but I would think the mass of anecdotal support for this exact position should suffice. I have a feeling you'll provide some flailing excuse.

Prove that algebra is incorporated into tacit knowledge or stop claiming that it is. Anecdotal support is not convincing in this instance, any more that the common wisdom of "if you keep your face that way it'll stick like that!"

5) On application. I am tired of fighting this false distinction you keep putting up. The skills of Algebraic reorganization and substitution are primarily concerned with the application of underlying mathematical concepts. This is why I have said that forgetting the application is the same as forgetting the skill, because they are skills of application, not algorithms or operations to be worked through but understanding how we can apply arithmetic rules. This is the thing you are seemingly unable to understand. Its like a physicist saying he knows the right hand rule for E&M but doesn't remember how to apply it or for a math example it is saying you know the rules of multiplication but don't know how to apply them. Could I say I know that multiplication is commutative if I don't realize I can switch the order of multiples involved? This is why people keeping using phrases like "really learned" to denote the difference between memorizing the words behind a concept and actually being able to understand the concept.

No, it's like someone knowing the right-hand-rule but forgetting that you use it in the case of rotation and electromagnetism, since we are talking about application. Someone might remember what factorization is, subconsciously, but forget that we apply it to n scenarios, or simply have forgotten the word "factorization" and be in the same scenario. Without some proof of the inability of people to forget algebra, you cannot declare that it is impossible for someone to forget how to apply the skill.

[Spoonist]

-Dark Helion
If you have not seen this then I believe that you will find this very interesting:
http://www.youtube.com/watch?v=WD1CXzTb ... =1&index=2
As Channel72 pointed out your whole excercise rests "on he shoulders of giants". Such things that are so simple to us took geniuses millenia to figure out.
So concepts like negative numbers, zero, power of, and especially multiple variables, these are not things that came easy to us humans.
So while I am also impressed by the effort you spent. It just shows that you are a product of our time, if you gave those dots and stuff to someone of our forefathers then they wouldn't be able to follow you. Because they would be conceptually challenged to follow the logic. So unless you provide them with a rosetta stone of sorts they would probably toss it away as incoherent.

Also go to an old folks home, there you can directly see (and test) what people forget first. You see things like motor skills and language are kept vastly longer than math skills, which deteriorate rapidly when you hit dementia. Even social skills like someone who has been taught how to use a "proper" posh table setting with three different forks etc, even that will be kept longer than math.

Another test you can do is simple fatigue tests. Like they do in the military. When you are at the limit of your physique you will not be able to perform even simple math, but you can ride a bike (if you learned how), fire a rifle and of course talk even with complex sentances. Drugs are similar depending on effect.

It does not trigger our fight/flight responses either, so we can not remember math through fear either. This so that stupid parents who use spanking as a teaching method will get the opposite effect when it comes to math, while not in social/motor skills. This phenomena is not fully understood but most likely an effect of how our brain functions under stress, cutting of less important stuff like what processes things like math.

Add to that the selective memory of positive/negative reinforcement. If you find math to be boring you are much more likely to forget specifics over time. Something which does not happen to the same extent with social/motor skills.

Math is not an inherent thing to us humans. It does not come to us naturally. Especially when you want people not only to memorize concepts and recite/use them but actually understand and be able to explain/prove the concept.

[Master of Ossus]

-Dark Helion
If you have not seen this then I believe that you will find this very interesting:
http://www.youtube.com/watch?v=WD1CXzTb ... =1&index=2
As Channel72 pointed out your whole excercise rests "on he shoulders of giants". Such things that are so simple to us took geniuses millenia to figure out.
So concepts like negative numbers, zero, power of, and especially multiple variables, these are not things that came easy to us humans.
So while I am also impressed by the effort you spent. It just shows that you are a product of our time, if you gave those dots and stuff to someone of our forefathers then they wouldn't be able to follow you. Because they would be conceptually challenged to follow the logic. So unless you provide them with a rosetta stone of sorts they would probably toss it away as incoherent.

Who cares? We're talking about the modern day. Anyone looking at this sort of math has obviously learned math as instructed by a series of instructors who all had the benefit of such knowledge. Moreover, you're exaggerating how hard it was to derive high school algebra. Even if it took a long time, almost everything taught in a high school class has been known for 2000 years.

Also go to an old folks home, there you can directly see (and test) what people forget first. You see things like motor skills and language are kept vastly longer than math skills, which deteriorate rapidly when you hit dementia. Even social skills like someone who has been taught how to use a "proper" posh table setting with three different forks etc, even that will be kept longer than math.

Another test you can do is simple fatigue tests. Like they do in the military. When you are at the limit of your physique you will not be able to perform even simple math, but you can ride a bike (if you learned how), fire a rifle and of course talk even with complex sentances. Drugs are similar depending on effect.

That was not my experience at all with fatigue tests, in which I retained my math skills far longer than I retained even the ability to speak in complex sentences and read with anything approaching comprehension and speed. I certainly did not lose my ability to do simple algebra!

It does not trigger our fight/flight responses either, so we can not remember math through fear either. This so that stupid parents who use spanking as a teaching method will get the opposite effect when it comes to math, while not in social/motor skills. This phenomena is not fully understood but most likely an effect of how our brain functions under stress, cutting of less important stuff like what processes things like math.

Add to that the selective memory of positive/negative reinforcement. If you find math to be boring you are much more likely to forget specifics over time. Something which does not happen to the same extent with social/motor skills.

Math is not an inherent thing to us humans. It does not come to us naturally. Especially when you want people not only to memorize concepts and recite/use them but actually understand and be able to explain/prove the concept.

Understanding is the only crucial thing. Once someone understands algebra, they will not forget it, and need not memorize it in order to use it to solve simple problems.

[Spoonist]

-MoO (and Dark Helion)
Ah, sorry, I should have been clearer. As I said I skipped most of Dark Helion and Bakustras middle arguments so I might have missed where their tangent started.
Completely conceded that given someone with a modern technical university education, who really understood the fundamentals, given enough time then they can figure out most formulas and such simply through deriving it from simpler stuff or trial and error. This because they may recall there is a formula but not exactly how it worked. So if that is what Dark Helion where talking about then I apologize.
But if that is the case then Dark Helion and Bakustra are talking about completely different things and are completely misunderstanding each other. In which case they should define their positions again.

Even if it took a long time, almost everything taught in a high school class has been known for 2000 years.

This is false, unless your "almost" is rather big. Well... It might also depend on if you are talking about the OP test alone???

That was not my experience at all with fatigue tests, in which I retained my math skills far longer than I retained even the ability to speak in complex sentences and read with anything approaching comprehension and speed

Really? Where do you draw the line for simple algebra? I found that everything that I had learned by rote memorization I could do like multiplication table and such stuff. Or simple square roots like 25, 36 etc. But as soon as I needed to "think" it through like the first five numbers of the square root of 14, or 88, etc I would be completely lost. Just like I could easily recite a2+b2=c2 but struggled to implement it.
So given the test in the OP I'd guesstimate that really fatigued I would not be able to solve, 1abc 2abd 3ab 4bcd, unless I could do it on paper and thereby get back some energy by stalling & breathing right.
I would probably solve 2c 4a because they would not require many steps.
While 5 & 6 would be too much info, which would get garbled.

Once someone understands algebra, they will not forget it, and need not memorize it in order to use it to solve simple problems.

If so I think our difference must lie in different definitions of algebra or "simple problems". Do you include abstract algebra? Linear algebra? Geometric algebra? Given the OP how much of that do you think is algebra?
Give some specific examples because you'd be amazed what "simple" math I have had to help engineering colleagues with, who simply forgot how to do it. Or better yet what kind of math do you think one can forget even if one "undestood" the underlying concept, because you seem to be stuck on 'simple algebra' for some reason.

Where do you put Quadratic equations? Most remember the solving formula by rote memorization, but struggle with actually implementing it to a problem unless it is exactly lined up as in the formula. This even if they understood the concept when in school.

[Ziggy Stardust]

Your ability to derive all of this from first principles is admirable, but if algebra is as intuitive as you claim then how come a human understanding of arithmetic existed for millenia before al Khwarizmi finally formalized the idea of combining like terms in a generic equation? Mankind has been capable of simple arithmetic for millenia, but only a few geniuses in the Hellenistic period ever considered algebraic concepts. It wasn't until the Middle Ages, after ~3,000 years of human civilization, that al Khwarizmi formalized the study of generic equations which eventually came to be called "algebra." So your expectation that the average person should be able to derive algebraic concepts from first principles based only on their knowledge of arithmetic is utter bullshit.

It is incredibly easy to make a fire. I can think of at least half a dozen ways to build one without effort. Do you know how long it took early humans to figure that out? Believe it or not, modern humans are smarter than ancient ones. News at 11.

Let's face it, the mathematical skill of the average human being is generally limited to basic arithmetic with natural numbers.

Proof?

Most pre-agricultural hunter-gatherer tribes didn't even have words in their language for numbers higher than 5, let alone zero or fractions. We may be in the 21st century, but the human brain hasn't changed.

The human brain has changed. The entire human body has changed. Humans, now, are smarter and taller and healthier than our pre-agricultural counterparts. Hell, most pre-agricultural hunter-gatherer tribes barely had complex languages at all.

[Rye]

It is incredibly easy to make a fire. I can think of at least half a dozen ways to build one without effort. Do you know how long it took early humans to figure that out? Believe it or not, modern humans are smarter than ancient ones. News at 11.

Smarter is a dodgy word to use. It implies that people are more intelligent now, and that's not the case. We're given more knowledge, rather than intelligence.

Proof?

Most of the planet is ill-educated.

The human brain has changed. The entire human body has changed. Humans, now, are smarter and taller and healthier than our pre-agricultural counterparts. Hell, most pre-agricultural hunter-gatherer tribes barely had complex languages at all.

How do you know this?

[ArmorPierce]

The human brain has changed. The entire human body has changed. Humans, now, are smarter and taller and healthier than our pre-agricultural counterparts. Hell, most pre-agricultural hunter-gatherer tribes barely had complex languages at all.

Well pre-agricultural hunter gatherer tribes tended to be relatively healthy and well fed. You wouldn't find a significant difference between their height and our own and I doubt any difference in brain processing power.

[Master of Ossus]

Smarter is a dodgy word to use. It implies that people are more intelligent now, and that's not the case. We're given more knowledge, rather than intelligence.

Well, there is also the Flynn effect going on.

The human brain has changed. The entire human body has changed. Humans, now, are smarter and taller and healthier than our pre-agricultural counterparts. Hell, most pre-agricultural hunter-gatherer tribes barely had complex languages at all.

Well pre-agricultural hunter gatherer tribes tended to be relatively healthy and well fed. You wouldn't find a significant difference between their height and our own and I doubt any difference in brain processing power.

Average hunter gatherer height was around 5'8". In the US it's 5'9.5" to 5'10", as of 2006.

[Master of Ossus]

Even if it took a long time, almost everything taught in a high school class has been known for 2000 years.

This is false, unless your "almost" is rather big. Well... It might also depend on if you are talking about the OP test alone???

How much, exactly, is covered in high school algebra? What wasn't known by the Greeks and Romans?

Really? Where do you draw the line for simple algebra? I found that everything that I had learned by rote memorization I could do like multiplication table and such stuff. Or simple square roots like 25, 36 etc. But as soon as I needed to "think" it through like the first five numbers of the square root of 14, or 88, etc I would be completely lost. Just like I could easily recite a2+b2=c2 but struggled to implement it.
So given the test in the OP I'd guesstimate that really fatigued I would not be able to solve, 1abc 2abd 3ab 4bcd, unless I could do it on paper and thereby get back some energy by stalling & breathing right.
I would probably solve 2c 4a because they would not require many steps.
While 5 & 6 would be too much info, which would get garbled.

I'm sorry, is your argument that rote memorization comes more easily to people than understanding underlying concepts? That seems difficult to believe, but consistent with your claims as to your performance during fatigue tests and your claims about what this indicates about the functions of the human brain.

If so I think our difference must lie in different definitions of algebra or "simple problems". Do you include abstract algebra? Linear algebra? Geometric algebra? Given the OP how much of that do you think is algebra?
Give some specific examples because you'd be amazed what "simple" math I have had to help engineering colleagues with, who simply forgot how to do it. Or better yet what kind of math do you think one can forget even if one "undestood" the underlying concept, because you seem to be stuck on 'simple algebra' for some reason.

I provided a California High School Exit Exam a little earlier in the thread. I would classify everything in it as "simple algebra," although I concede that I haven't thought too deeply about the boundaries of the distinction beyond that.

Where do you put Quadratic equations? Most remember the solving formula by rote memorization, but struggle with actually implementing it to a problem unless it is exactly lined up as in the formula. This even if they understood the concept when in school.

Oh, come on. If you remember the formula you ought to be able to apply it. Otherwise you simply haven't learned the formula.

[ArmorPierce]

Smarter is a dodgy word to use. It implies that people are more intelligent now, and that's not the case. We're given more knowledge, rather than intelligence.

Well, there is also the Flynn effect going on.

The human brain has changed. The entire human body has changed. Humans, now, are smarter and taller and healthier than our pre-agricultural counterparts. Hell, most pre-agricultural hunter-gatherer tribes barely had complex languages at all.

Well pre-agricultural hunter gatherer tribes tended to be relatively healthy and well fed. You wouldn't find a significant difference between their height and our own and I doubt any difference in brain processing power.

Average hunter gatherer height was around 5'8". In the US it's 5'9.5" to 5'10", as of 2006.

I don't think that is a significant difference, do you? In fact, I've seen numbers putting the average height at higher, at about 5'10" for hunter gatherers (I've seen 5'8" too though). During the agricultural revolution, the average male height went to down to 5'5"-5'6".

[Master of Ossus]

I don't think that is a significant difference, do you?

Yes, I do. These are off-shifted bell curves; a 2-inch difference in the mean is significant. Moreover, how do you consider a drop of 2-4 inches (from 5'8" to 5'6") significant if an increase from 5'8" to 5'10" is not?

[Spoonist]

I'll seperate my answers for clarity. First a short one on the weird height tangent.

Most pre-agricultural hunter-gatherer tribes didn't even have words in their language for numbers higher than 5, let alone zero or fractions. We may be in the 21st century, but the human brain hasn't changed.

The human brain has changed. The entire human body has changed. Humans, now, are smarter and taller and healthier than our pre-agricultural counterparts. Hell, most pre-agricultural hunter-gatherer tribes barely had complex languages at all.

Yes, I do. These are off-shifted bell curves; a 2-inch difference in the mean is significant. Moreover, how do you consider a drop of 2-4 inches (from 5'8" to 5'6") significant if an increase from 5'8" to 5'10" is not?

The average Japanese male has increased from 157.9 cm in 1900 to 170.8 cm in 2000. This due to a change in food consumption from mostly agricultural products like rice to a more rich one with more animal products. (I could give other examples from modern or medieval europe if you wish).
To claim that "the japanese brain has changed" because of this would be literally true but not genetically so. Yes, with the height change comes a weight change so japanese brains are bigger than they where a 100 years ago. This could be argued to lead to a change in intelligence as well because we know that starvation and vitamin deficiances lead to lowered brain function. But genetically? Almost no change at all.
But to extrapolate stuff like that to "the human brain has changed" would be missing the context. Genetically we have not changed as much as we have culturally. Of course we are smarter now than we used to be, but that is because of education in several generations. Not a change in brain function.

That would be a complete failure of understanding genetic evolution.

So if MoO & Ziggy Stardust is talking about the cultural & economic change then yes we are smarter than our forefathers. If they are talking about genetic/DNA then they are proof that we are not.

[ArmorPierce]

I don't think that is a significant difference, do you?

Yes, I do. These are off-shifted bell curves; a 2-inch difference in the mean is significant. Moreover, how do you consider a drop of 2-4 inches (from 5'8" to 5'6") significant if an increase from 5'8" to 5'10" is not?

I would guess what we consider to be 'significant' would be relative to something else. If we accept the average modern height of 5'10" for an American and the low range for the pre-agricultural human at 5'8" (as I've stated, I've seen each numbers in between 5'8" and 5'10" stated), 5'8" falls within a standard deviation of the average height of a modern man whilst 5'6" falls outside of that (correct me if I'm wrong). It would be a 2.9% difference vs a 5.7% difference in height. Anyway, I didn't say any was significant. Note that 5'8" is still higher than the world average of height which is at around 5'6".

[Spoonist]

MoO I think that these paralell discussions are really hurting our understanding of each other. You have now responded again with a high school reference directly after quoting me talking about "math"/"algebra" in general terms or engineer level. If you all this time has been only thinking high school math in your head when responding in this topic then I think that you have missed several points completely.

Regarding the mil fatigue I'm genuinly curious how our experiences could be so different. So if you don't mind could you elaborate a bit on what kind of math tests you had and where you think that your limit to math tests would go when fatigued. You said you retained your math skills, up to which level would that be? How complex problems did you solve? Because I'm good at math and I had some big fucking problems and so did my whole troop of roughly 80 people who where of mixed math profficiency.

I'm sorry, is your argument that rote memorization comes more easily to people than understanding underlying concepts? That seems difficult to believe...

Under heavy stress, hell yes. That is the reason why we have repetetive drills in the military. This so that regardles of how tired or stressed you are, you should still be able to perform X action. In most skill based learning this is the basics. Learning a new language you do it with word vocabularies or reciting irregular verbs. In math its tables and formulas.
How you could be ignorant of this could only be explained by you having a different definition of rote memorization than I do.
Mine is that you repeat something until it "sticks". What is yours? (I also checked wiki so it wouldn't be a cultural/language problem, and wiki had pretty much the same definition we do).

I provided a California High School Exit Exam a little earlier in the thread. I would classify everything in it as "simple algebra," although I concede that I haven't thought too deeply about the boundaries of the distinction beyond that.

So no example of what you think can be forgot regardless of understanding it? Is this because you think there is no limit, that as long as you have understood a math concept you will forever be able to solve such problems? I hope that is not the case.

Oh, come on. If you remember the formula you ought to be able to apply it. Otherwise you simply haven't learned the formula.

That is bullshit. What kind of reality do you live in? By that "logic" people wouldn't fail math tests if they know the formula. Something which people consistently do. Regardless if its high school, college, uni, or whatever.

Why do you think its more difficult for people to do the text questions on math tests than the ones with only numbers? That's because even if they do remember the formula they fail to apply it correctly or even do not realise which formula it indicates. Same thing applies with this.
People, even though remembering the quadratic equation formula, sometimes do not recognize that is what would solve a problem/question. Especially if it was years since they last used it.

[Korgeta]

I manage to get upto 5 and called it a day, for a 14yr old paper it was a hard one (Though this seems more of a higher paper than a foundation one) as I see this question more in the higher papers whilst working as Exam Invigilator.

I have always struggled with maths for a completely different reason, a lot of the teachers felt i was lazy, falling asleep a lot in class, however it took a recommendation to see a different GP by my driving instructor who believed I had glaucoma which was responsible for my poor focus where my sight is more or less at a stage where it's double vision on numbers and spelling. Turned out he was right, though by the time it was picked up school was long done, however after receiving advice, medication and better glasses I have been able to catch up (gradually on what I've missed) It also helped that am doing GCSE Maths foundation (at evening class) and key skills level 2 which is equivalent to a grade C. However key skills is problem solving with multiplication, fractions and division. There is no algebra in it at all, it's all logic solving. Whereas the GCSE is harder and does include algebra. I'll be doing exams this year and hopefully next year work onto A'S level.

so no I can't do grade 9 maths fully but one day maybe I will be able to.